APPLICATION OF NEURAL NETWORK TO SHAPE OPTIMIZATION PROBLEM

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1 Aliguliyev R.M., Majidzade K., Gasei Habasi Y. Institute of Inforation Tecnology AAS, Baku, Azerbaan Institute of Applied Mateatics Baku State University APPLICATIO OF EURAL ETWORK TO SHAPE OPTIMIZATIO PROBLEM In te present paper we consider sape optiization proble and reduce it to te integer prograing by discretization of te original proble. Tis forulation of te considered proble allows applying te neural networks for its solving. Keywords: sape optiization, artificial neural network, approxiate teory, network training.. Introduction Identification of non-linear systes is an iportant task for odel based control, syste design, siulation, prediction and fault diagnosis. For investigating tis kind of question or proble, we can use neural networks. In recent years artificial neural networks ave gained a wide attention in control applications. It is te ability of te artificial neural networks to odel non-linear systes tat can be te ost readily exploited in te syntesis of non-linear controllers. Artificial neural networks ave been used to forulate a variety of control strategies [, ]. In present paper, we consider iniization of te doain dependant integral functional. But since suc type probles are sape optiization probles, teir investigation is connected wit soe ateatical difficulties [3 7]. Tis difficulty is related ainly to te ateatical definition of te variation of te doain caracterized by its boundary variation. Te new approac introduced in [5 7] tends to avoid tese difficulties. It consists of representing a convex doain by its support function. Te variation of te doain ten is naturally replaced by te variation of te corresponding support function. In te nuerical siulation process after eac iteration we get not only a set of boundary points, but also a support function. Te doain is reconstructed as a sub-differential of its support function at te point 0 [8]. We see tis tecnique allowed to solve suc type probles wen te considered doains are convex. But, convexity of te doains is a very ard condition fro te practical point of view. Terefore, in order to solve tis proble we reduce it to te integer prograing proble by its discretization. We applied a neural network for solving te integer prograing proble.. Stateent of te proble By K we denote a set of doains D R. K ay be deterined for exaple, by fixing te area, te lengt of te boundary or by te relation D0 D D and etc. Here D, D R 0 are te given doains. Suppose tat tere exists a rectangle Q suc tat if D K, ten D Q. We consider te following functional Here, (x) = D J ( D) f ( x) dx () f is a function continuous on Q. If f ( x), x Q, ten J ( D) = esd. 47

2 Let it be required to iniize te functional () on te set K. For solving of tis proble we discretize te rectangle Q wit respect to unifor sall step > 0. We denote te obtained lattice by Q. 3. Discretization and solving of te proble It is clear tat in tis case we can oppose to eac D Q its discrete analogy Ω Q lattice. By d, i =,, j =, we denote te square of te lattice Q corresponding to te i t row and j t colun. Here and are te nubers of vertical and orizontal partition points, respectively. ow, we discretize te functional (). We denote, x D H D ( x) =. 0, x D Ten, it is clear tat J ( D) = f ( x) H ( x) dx. () We introduce te following denotation, d Ω z =. 0, d Ω Ten we can discretize functional () as follows Q i= j= D I = f z (3) Here, f are te values of te function f (x) at any point of te square d Q. In tis case, we can give te discrete analogy of te restriction Ω K. For exaple, if te section K is a set of doains wose area equal te nuber c, ten its discrete analogy consists of te set of lattices Ω satisfying te condition c z = (4) i= j= c Here we ll assue tat is suc a step tat te nuber c = is natural. In te sae way, te oter restrictions also ay be written in te discrete for. For definiteness, suppose tat te restricition K is as esd = c. Tus, te proble on iniization of functional () provided D K is reduced to integer prograing proble i= j= i = j = f in (5) z z = c (6) z {0,} (7) To solve te integer prograing proble (5) (7) we can use te MATLAB progra packets. Assue tat we ave solved proble (5) (7) and found te variables z, i =,, j =,. Ten te approxiately solved lattice Ω is defined as follows: 48

3 Ω = d : z =, i =,, j, } (8) { = Reduction of iniization of doain dependent functional () to proble (5)-(7) allows to apply te artificial neural network to solve tis proble. It is seen, aving given te diensional atrix F = ( f ), i =,, j =,, we get te atrix Z = ( z ) as a solution of proble (5) (7). In te solution of proble (5) (7) atrix Z canges due to cange of te atrix F. So te lattice Ω giving iniu to functional () also canges. Using tis fact we give te scee of application of te artificial neural network to solve te proble on iniization of functional ().. Te proble on iniization of functional provided D K () is reduced to integer prograing proble (5) (7).. Having solved proble (5) (7) for nuber atrices F = ( f ), F = ( f ),, F = ( ) we find te atrices Z = ( z ), Z = ( z ),, Z = ( ). f z 3. We ust create suc an artificial neural network tat it could associate te output atrices Z, Z,, Z to input atrices F, F,, F. Assue tat te artificial neural network corresponding to tese input and output as been establised. Denote tis artificial neural network by C (). 4. Assue tat our ai it is find te atrix Z = z ) being te solution of proble (5)-(7) according to te given atrix F = ( f ), i =,, j =,. For tat we give te atrix F = f ) as input variables of te constructed artificial neural network ( C (). As an output variable tis lattice C () will give us a new atrix Z = ( z ). We ll accept tis atrix as an approxiate solution of proble (5)-(7). 5. Te lattice Ω corresponding to te atrix Z = z ) is establised by forula (8). otice tat te establisent of te lattice corresponding to te atrix ay be realized by te application of te artificial neural network. otice tat as te nuber of te input atrices increases, te exactness of an approxiate solution deterined by te artificial neural network will be also iproved. ow, using tis scee, we solve proble () on soe odel exaple and analyze te obtained results. x x Exaple. Let f ( x) = +. In tis case solution of te proble () is ellipse 6 4 wit axes a = 4, b =. Applying te considered algorit, we obtain =5 (figure) and =5 (figure ). Tus, applying a neural network we can solve te sape optiization proble approxiately and as seen tis solution is close to te exact solution. ( ( 49

4 İnforasiya texnologiyaları probleləri, problel, 00 Figure. =5 Figure. =5 4. Conclusion Using new approac sape optiization proble reduced to integer prograing proble. Tis allowed applying neural networks to solve te proble. We considered soe odel exaples wic sow te effectiveness of te proposed approac. References. A.U.Levin and K.S.arendra Control of nonlinear dynaical systes using neural networks: controllability and stabilization stabilization // IEEE Transactions on eural etworks, 993, no.4, pp pp.9. R.M.Alguliev, R.M.Aliguliyev and R.K.Alekperov, An approac to optial task assignent in a distributed syste // Journal Autoation and Inforation science, 004, v.36, no.0, pp pp.5 3. J.Sea, uerical etod in ateatical pysics andd optial control. ovosibirsk, M.: auka, 978, 40p. 4. J.Sokolowski and J.-P.Zolesio P.Zolesio, Introduction oduction to sape optiization. Sape sensitivity analysis, Springer, Heidelberg, 99, 585p. 5. A.A.iftiyev and E.R.Akadov.R.Akadov, Variational stateent of an inverse proble for a doain // Journal of Differential Equation, 007, v.43, no.0, pp pp Y.S.Gasiov and A.A.iftiyev, A.A.iftiyev On a iniization of te eigenvalues of Scrödinger operator over doains // Doklady RAS,, 00, v.380, no.3, pp pp A.A.iftiyev and Y.S.Gasiov, Y.S.Gasiov Control by boundaries and eigenvalue probles by variable doain. Baku: BSU, 004, 85p. 8. V.F.Deyanov and A.M.Rubinov, Bases of non-soot soot analysis and quasidifferential calculus. M.: auka, auk 990, 4p. 9. F.P.Vasilyev, uerical etods of solution of te extree probles. probles M.: auka, 980, 58 p. 0. D.M.Burago and V.A.Zalgaller, V.A.Zalgaller Geoetrical ical inequalities. M.: auka, 98, 400p.. E.R.Aadov, Discrete optial control proble wit unknown points poin // Transaction of AAS, Inforatics and Control Probles, 006, v.xxvi, v. no., pp

5 UOT 57.97; Alıquliyev R.M., Məсidzadə K., Qəsei Həbəşi Y. AMEA İnforasiya Texnologiyaları İnstitutu, Bakı, Azərbaycan Bakı Dövlət Universitetinin Tətbiqi Riyaziyyat Eli-Tədqiqat İnstitutu Foraya görə optiallaşa əsələsinə neyron şəbəkənin tətbiqi Məqalədə optial foranın tapılası əsələsinə baxılır və əsələ diskretləşdirilərək ta qiyətli optiallaşa əsələsinə gətirilir. Məsələnin belə qoyuluşu onun əllinə neyron şəbəkənin tətbiqinə ikan verir. Açar sözlər: fora optiallaşa, süni neyron şəbəkələr, aproksiasiya nəzəriyyəsi, şəbəkənin öyrədiləsi. УДК 57.97; Алыгулиев Р.М., Маджид-заде К., Гасеми Хабаши Я. Институт Информационных Технологий НАНА, Баку, Азербайджан Научно-Исследовательский Институт Прикладной Математики при БГУ Применение нейронной сети к задаче оптимизации по форме В настоящей работе рассматривается задача нахождения оптимальной формы, и она с дискретизацией сводится к задаче целочисленного программирования. Такой подход позволяет применить нейронные сети для решения поставленной задачи. Ключевые слова: оптимизация формы, искусственные нейронные сети, теория аппроксимации, обучение сетей. 5